The geometry of the Hausdorff domain in localization problems for the spectrum of arbitrary matrices

In this article it is shown that the Hausdorff domain (numerical range) $W(A)=\{(Ax,x):\|x\|=1\}$ is the union of the numerical ranges of a concretely constructed family of matrices acting in $\mathbf C^2$. In other words, a certain method of descent of the numerical range is justified. This method is used to study localizations for the spectra of arbitrary matrices. As a result, generalizations are discovered for results of Johnson, Gershgorin–Solov'ev, Hirsch and Bendixson, and Mees and Atherton.
Bibliography: 20 titles.